On the L^p solutions of dilation equations

Let A ∈ M_n(ℤ) be an expanding matrix with |det(A)| = q and let K = {k₁ ⋯ k_q} ⊆ ℝⁿ be a digit set. The set T =: T(A, K) = {∑_i=1^∞ A^-i k_ji : k_ji ∈ K} ⊂ ℝⁿ is called a self-affine tile if the Lebesgue measure of T is positive. In this note, we consider dilation equations of the form f(x) = ∑_j=1^q c_jf(Ax - k_j) with q = ∑_j=1^q c_j, c_j ∈ ℝ, and prove that this equation has a nontrivial L_p solution (1 ≤ p ≤ ∞) if and only if c_j = 1 ∀j ∈ {1,...,q} and T is a tile.